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Theorem ssrabdv 3100
Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)
Hypotheses
Ref Expression
ssrabdv.1  |-  ( ph  ->  B  C_  A )
ssrabdv.2  |-  ( (
ph  /\  x  e.  B )  ->  ps )
Assertion
Ref Expression
ssrabdv  |-  ( ph  ->  B  C_  { x  e.  A  |  ps } )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem ssrabdv
StepHypRef Expression
1 ssrabdv.1 . 2  |-  ( ph  ->  B  C_  A )
2 ssrabdv.2 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ps )
32ralrimiva 2446 . 2  |-  ( ph  ->  A. x  e.  B  ps )
4 ssrab 3099 . 2  |-  ( B 
C_  { x  e.  A  |  ps }  <->  ( B  C_  A  /\  A. x  e.  B  ps ) )
51, 3, 4sylanbrc 408 1  |-  ( ph  ->  B  C_  { x  e.  A  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1438   A.wral 2359   {crab 2363    C_ wss 2999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-in 3005  df-ss 3012
This theorem is referenced by: (None)
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